Let $R$ be a commutative ring; does every ideal in the $I$-adic completion of $R$
$$ \varprojlim_i R/I^i $$
arise as the $I$-adic completion of some ideal inside of the original ring $R$?
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Sign up to join this communityLet $R$ be a commutative ring; does every ideal in the $I$-adic completion of $R$
$$ \varprojlim_i R/I^i $$
arise as the $I$-adic completion of some ideal inside of the original ring $R$?
No: take the ring $k[[t_1, ..., t_n, ...]]$ of power series of countably many variables, and let $I$ be the ideal generated by $r:=\sum t_i^i$. Since there is no invertible $u$ such that $r=uP$, where $P$ is a polynomial, the ideal $I$ does not come from any ideal in the polynomial ring.
Since irreducibility of noetherian schemes is not local for the etale topology, one gets zillions of noetherian counterexamples from irreducible varieties which are not "analytically irreducible". To be specific, let $R$ be any noetherian local domain whose completion has reducible spectrum (e.g., $R = k[x,y]/(y^2 = x^2(x+1))$ for any field $k$ not of characteristic 2) and let $I$ be the maximal ideal. Then every minimal prime of $\widehat{R}$ lies over $(0)$ in $R$ due to faithful flatness of such completion, so no such prime can arise from $R$ as such primes are nonzero (since there is more than one of them).